• Corpus ID: 239015942

Finite Sections of Periodic Schrödinger Operators

  title={Finite Sections of Periodic Schr{\"o}dinger Operators},
  author={Fabian Gabel and Dennis Gallaun and Julian Grossmann and Marko Lindner and Riko Ukena},
We study discrete Schrödinger operators H with periodic potentials as they are typically used to approximate aperiodic Schrödinger operators like the Fibonacci Hamiltonian. We prove an efficient test for applicability of the finite section method, a procedure that approximates H by growing finite square submatrices Hn. For integer-valued potentials, we show that the finite section method is applicable as soon as H is invertible. This statement remains true for {0, λ}-valued potentials with… 

Figures and Tables from this paper


Spectra of discrete two-dimensional periodic Schrödinger operators with small potentials
We show that the spectrum of a discrete two-dimensional periodic Schrodinger operator on a square lattice with a sufficiently small potential is an interval, provided the period is odd in at least
Spectral Approximation for Quasiperiodic Jacobi Operators
Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals and in more general studies of structures exhibiting aperiodic order. The spectra of these self-adjoint operators can be
Finite sections of the Fibonacci Hamiltonian
We study finite but growing principal square submatrices An of the one- or two-sided infinite Fibonacci Hamiltonian A. Our results show that such a sequence (An), no matter how the points of
Spectral analysis of tridiagonal Fibonacci Hamiltonians
We consider a family of discrete Jacobi operators on the one-dimensional integer lattice, with the diagonal and the off-diagonal entries given by two sequences generated by the Fibonacci substitution
Jacobi Operators and Completely Integrable Nonlinear Lattices
Jacobi operators: Jacobi operators Foundations of spectral theory for Jacobi operators Qualitative theory of spectra Oscillation theory Random Jacobi operators Trace formulas Jacobi operators with
The Fibonacci Hamiltonian
We consider the Fibonacci Hamiltonian, the central model in the study of electronic properties of one-dimensional quasicrystals, and establish relations between its spectrum and spectral
The finite section method and stable subsequences
Finite sections of band-dominated operators
In an earlier paper we showed that the sequence of the finite sections P n AP n of a band-dominated operator A on l p (ℤ) is stable if and only if the operator A is invertible, every limit operator
Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices
In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner
Fredholm theory and finite section method for band-dominated operators
The topics of this paper are Fredholm properties and the applicability of the finite section method for band operators onlp-spaces as well as for their norm limits which we call band-dominated